Category Theory and Foundations

2018 ◽  
Author(s):  
Michael Ernst
Keyword(s):  
Category Theory ◽  
Mathematical Thinking ◽  
Mathematical Practice ◽  
Foundations Of Mathematics ◽  
Ongoing Debate ◽  
Technical Adequacy ◽  
Theoretical Foundations

In the foundations of mathematics there has been an ongoing debate about whether categorical foundations can replace set-theoretical foundations. The primary goal of this chapter is to provide a condensed summary of that debate. It addresses the two primary points of contention: technical adequacy and autonomy. Finally, it calls attention to a neglected feature of the debate, the claim that categorical foundations are more natural and readily useable, and how deeper investigation of that claim could prove fruitful for our understanding of mathematical thinking and mathematical practice.

Supporting Preservice Secondary Mathematics Teachers’ Professional Judgment Around Digital Technology Use

2021 ◽  
Vol 9 (2) ◽  
pp. 145-159
Author(s):  
Charmaine Mangram ◽  
Kathy Liu Sun
Keyword(s):  
Preservice Teachers ◽  
Digital Technology ◽  
Technology Use ◽  
Teacher Educators ◽  
Mathematical Thinking ◽  
Mathematical Practice ◽  
Secondary Mathematics ◽  
Ease Of Use ◽  
Professional Judgment ◽  
Mathematics Methods

The pervasiveness of digital technology creates an imperative for mathematics teacher educators to prepare preservice teachers (PSTs) to select technology to support students’ mathematical development. We report on research conducted on an assignment created for and implemented in secondary mathematics methods courses requiring PSTs to select and evaluate digital mathematics tools. We found that PSTs primarily focused on pedagogical fidelity (ease of use), did not consider mathematical fidelity (accuracy), and at times superficially attended to cognitive fidelity (how well the tool reflects students’ mathematical thinking processes) operationalized as the CCSS for Mathematical Practice and Five Strands of Mathematical Proficiency. We discuss implications for implementing the assignment and suggestions for addressing PSTs’ challenges with identifying the mathematical practices and five strands.

Category theory, applications to the foundations of mathematics

2018 ◽  
Author(s):  
Colin McLarty
Keyword(s):  
Set Theory ◽  
Category Theory ◽  
Classical Logic ◽  
Elementary Theory ◽  
Mathematical Structure ◽  
Point Of View ◽  
Foundations Of Mathematics ◽  
Logical Foundations ◽  
Classical Mathematics ◽  
The 1960S

Since the 1960s Lawvere has distinguished two senses of the foundations of mathematics. Logical foundations use formal axioms to organize the subject. The other sense aims to survey ‘what is universal in mathematics’. The ontology of mathematics is a third, related issue. Moderately categorical foundations use sets as axiomatized by the elementary theory of the category of sets (ETCS) rather than Zermelo–Fraenkel set theory (ZF). This claims to make set theory conceptually more like the rest of mathematics than ZF is. And it suggests that sets are not ‘made of’ anything determinate; they only have determinate functional relations to one another. The ZF and ETCS axioms both support classical mathematics. Other categories have also been offered as logical foundations. The ‘category of categories’ takes categories and functors as fundamental. The ‘free topos’ (see Lambek and Couture 1991) stresses provability. These and others are certainly formally adequate. The question is how far they illuminate the most universal aspects of current mathematics. Radically categorical foundations say mathematics has no one starting point; each mathematical structure exists in its own right and can be described intrinsically. The most flexible way to do this to date is categorically. From this point of view various structures have their own logic. Sets have classical logic, or rather the topos Set has classical logic. But differential manifolds, for instance, fit neatly into a topos Spaces with nonclassical logic. This view urges a broader practice of mathematics than classical. This article assumes knowledge of category theory on the level of Category theory, introduction to §1.

The maclane problem on set-theoretical foundations for the category theory

2000 ◽  
Vol 102 (3) ◽  
pp. 4018-4031
Author(s):  
V. K. Zakharov ◽  
A. V. Mikhalev
Keyword(s):  
Category Theory ◽  
Theoretical Foundations

CATEGORICAL FOUNDATIONS OF MATHEMATICS OR HOW TO PROVIDE FOUNDATIONS FORABSTRACTMATHEMATICS

2012 ◽  
Vol 6 (1) ◽  
pp. 51-75 ◽  
Author(s):  
JEAN-PIERRE MARQUIS
Keyword(s):  
Category Theory ◽  
The Other ◽  
Foundations Of Mathematics ◽  
The Way

AbstractFeferman’s argument presented in 1977 seemed to block any possibility for category theory to become a serious contender in the foundational game. According to Feferman, two obstacles stand in the way: one logical and the other psychological. We address both obstacles in this paper, arguing that although Feferman’s argument is indeed convincing in a certain context, it can be dissolved entirely by modifying the context appropriately.

Rigor and Clarity: Foundations of Mathematics in France and England, 1800–1840

1991 ◽  
Vol 4 (2) ◽  
pp. 297-319 ◽  
Author(s):  
Joan L. Richards
Keyword(s):  
Nineteenth Century ◽  
Mathematical Practice ◽  
Fundamental Question ◽  
Foundations Of Mathematics ◽  
The Real ◽  
Turn On ◽  
The Subject ◽  
Diverse Groups

The ArgumentIt has long been apparent that in the nineteenth century, mathematics in France and England developed along different lines. The differences, which might well be labelled stylistic, are most easy to see on the foundational level. At first this may seem surprising because it is such a fundamental area, but, upon reflection, it is to be expected. Ultimately discussions about the foundations of mathematics turn on views about what mathematics is, and this is a question which is answered by a variety of different groups including mathematicians, students, curricular planners, parents, etc. Mathematical practice rests on some kind of mixture of the answers to this fundamental question which come from these diverse groups. Comparing the cultural matrices which supported mathematics in France and Britain in the first decades of the nineteenth century sheds light on the real though often subtle differences in the ways the subject was pursued in the two countries.

NEITHER CATEGORICAL NOR SET-THEORETIC FOUNDATIONS

2012 ◽  
Vol 6 (1) ◽  
pp. 16-23
Author(s):  
GEOFFREY HELLMAN
Keyword(s):  
Modal Logic ◽  
Set Theory ◽  
Category Theory ◽  
Top Down ◽  
Ongoing Debate ◽  
Modest Proposal ◽  
Categories And Functors ◽  
Theoretic Foundations

AbstractFirst we review highlights of the ongoing debate about foundations of category theory, beginning with Feferman’s important article of 1977, then moving to our own paper of 2003, contrasting replies by McLarty and Awodey, and our own rejoinders to them. Then we offer a modest proposal for reformulating a theory of category of categories that would actually meet the objections of Feferman and Hellman and provide a genuine alternative to set theory as a foundation for mathematics. This proposal is more modest than that of our (2003) in omitting modal logic and in permitting a more “top-down” approach, where particular categories and functors need not be explicitly defined. Possible reasons for resisting the proposal are offered and countered. The upshot is to sustain a pluralism of foundations along lines actually foreseen by Feferman (1977), something that should be welcomed as a way of resolving this long-standing debate.

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